Charting nearby dust clouds using Gaia data only
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Astronomy & Astrophysics manuscript no. paper c ESO 2019
May 16, 2019
Charting nearby dust clouds using Gaia data only
R. H. Leike12 and T. A. Enßlin12
1
Max Planck Institute for Astrophysics, Karl-Schwarzschildstraße 1, 85748 Garching, Germany
2
Ludwig-Maximilians-Universität, Geschwister-Scholl Platz 1, 80539 Munich, Germany
Received XXXX, accepted XXXX
ABSTRACT
arXiv:1901.05971v2 [astro-ph.GA] 15 May 2019
Aims. Highly resolved maps of the local Galactic dust are an important ingredient for sky emission models. In nearly the whole
electromagnetic spectrum one can see imprints of dust, many of which originate from dust clouds within 300pc. Having a detailed 3D
reconstruction of these local dust clouds enables detailed studies, helps to quantify the impact on other observables and is a milestone
necessary to enable larger reconstructions, as every sightline for more distant objects will pass through the local dust.
Methods. To infer the dust density we use parallax and extinction estimates published by the Gaia collaboration in their second data
release. We model the dust as a log-normal process using a hierarchical Bayesian model. We also infer non-parametrically the kernel
of the log-normal process, which corresponds to the physical spatial correlation power spectrum of the log-density.
Results. Using only Gaia data of the second Gaia data release, we reconstruct the 3D dust density and its spatial correlation spec-
trum in a 600pc cube centered on the Sun. We report a spectral index of the logarithmic dust density of 3.1 on Fourier scales with
wavelengths between 2pc and 125pc. The resulting 3D dust map as well as the power spectrum and posterior samples are publicly
available for download.
Key words. ISM: dust, extinction – Galaxy: local interstellar matter – methods: data analysis
1. Introduction driving force for 3D dust reconstruction and astronomy in gen-
eral are large surveys like 2MASS (Skrutskie et al. 2006), Pan-
Emission and extinction by Galactic dust is a prominent astro- STARRS (Kaiser et al. 2002) and SDSS/APOGEE (Albareti
nomical foreground at many wavelengths. Therefore, knowing et al. 2017) and WISE (Wright et al. 2010). These surveys pro-
its distribution on the 2D sky and in 3D is essential for many vide photometric measurements and some of them spectra for
astronomical observations. However, dust is also interesting to thousands of stars, from which the calculation of photometric
be studied on its own, as it provides information about the phys- distances is possible. There are two 3D dust reconstructions
ical conditions in the interstellar medium and informs us about based on these data sets that are closely related to the approach
star forming regions. Dust has been mapped out by surveys for a taken in this paper.
long time, the first notable contribution being Burstein & Heiles
(1978). Their dust reconstruction, as most reconstructions of the In Lallement et al. (2018) reddening data from 71 000
dust distribution so far, was focused on mapping the dust in 2D sources has been used to perform a 3D reconstruction using
on the sky. This can be done by looking at the sky in infra-red Gaussian process regression. The resulting dust map covers a
wavelengths, where it is dominated by dust emission. However, 4kpc square of the galactic plane and 600pc in perpendicular di-
when mapping out dust using infra-red emission one is biased rection with a voxel size of (5 pc)3 .
by the radiation field, as dust emission is not only proportional In Green et al. (2018) a 3D dust map is produced by com-
to the dust density but also to the amount of starlight absorbed bining the star data of Pan-STARRS and 2MASS, binning it in
by the dust. Furthermore, dust maps that were produced by map- angular and distance bins, and performing independent Bayesian
ping infrared light might contain extended infrared sources that reconstructions per angular bin. The result is a dust map that
are not within our galaxy, as was shown by Chiang & Ménard covers three quarters of the sky to a distance up to 2kpc. This
(2018). On the other hand, a hypothetical cold dust cloud cannot reconstruction shows artificial radial structures called the "fin-
be seen in infra-red, leading to systematic errors in the analysis gers of God effect" in analogy to the well known phenomenon
of distant targets, for example quasars or the Cosmic Microwave in cosmology (Jackson 1972; Tully & Fisher 1978). One way to
Background (CMB). mitigate this effect is to use more accurate parallax information.
For accurate analyses of objects in our galactic vicinity, it A prominent new survey is performed by the Gaia collabora-
is vital to have a 3D dust map as a foreground model, which tion (Brown et al. 2016). In its second data release (DR2, Brown
informs us about regions that cannot be observed, or only be et al. (2018)) accurate parallaxes for roughly 2 billion stars are
observed with less fidelity, due to dust obscuration. The first published. The provided catalogue also contains estimates of ex-
non-parametric reconstruction of galactic dust in 3D published is tinction coefficients for a subset of about 88 million stars, using
Arenou et al. (1992). Since then there have been many attempts spectral information of the Gaia satellite’s three energy bands.
to chart the dust density in 3D in increasing resolution, accuracy Due to the limited spectral information, the accuracy of the ex-
and for an ever greater part of our galaxy (Vergely et al. 1997, tinction coefficients estimated for individual sightlines is quite
2001; Chen et al. 2018; Kh et al. 2017, 2018a,b; Gontcharov low. For this reason it is recommended by Andrae et al. (2018) to
2012, 2017; Sale et al. 2014; Sale & Magorrian 2018). A large not use the information of individual sightlines but only the joint
Article number, page 1 of 19A&A proofs: manuscript no. paper
information of several sightlines. Even though the data quality
of individual sightlines is rather low, the sheer amount of data
points and the accuracy of the parallaxes outweigh this limita-
tion as our work shows.
So far, 3D dust reconstructions have never been performed
using solely Gaia data, instead Gaia data has been used for its ac-
curate parallax measurements only and the more accurate spec-
tral information of other surveys was used (Lallement et al. 2018,
for example).
In this paper we present a 3D dust reconstruction us-
ing Gaia DR2 data only. The results of the reconstruction
are provided online on https://wwwmpa.mpa-garching.
mpg.de/~ensslin/research/data/dust.html or by its
doi:10.5281/zenodo.2577337, and can be used under the terms
of the ODC-By 1.0 license. The inference of the unknown dust
density from the extinction data is performed by a critical filter, 2 3 4 5 6 7
a method for Gaussian process regression with non-parametric
kernel learning, first published in Enßlin & Frommert (2011).
While the statistical model used here is up to minor details equiv-
alent to the model introduced in that paper, the algorithm to ar-
rive at an approximate posterior summary statistics is quite dif-
ferent. The relevant numerical method used in this paper is out-
lined in Knollmüller & Enßlin (2019), which describes a general
method to derive posterior summary statistics for high dimen-
sional Bayesian inference problems. For a theoretical discussion
of the underlying inference framework of information field the-
ory we refer to Enßlin (2018). The algorithm was implemented
using the Python package NIFTy5, which is the newest version
of the software package NIFTy (Selig 2013; Steininger et al.
2017; Arras et al. 2019a)1 . Even though mathematical theory,
statistical motivations, and numerical details are distributed over
the aforementioned papers, this paper is entirely self-contained
−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0
by describing the whole method.
In Sec. 2 we discuss which part of the Gaia data we used. We
introduce our statistical model of the interstellar dust density as
well as of the measurement in Sec. 3. In Sec. 4 we present a test
application of the algorithm using synthetic data, verifying the
predictive power of the algorithm. The main results of the dust
density reconstruction using Gaia data are presented in Sec. 5.
This section also contains a brief recommendation on how to
use our results. Our dust reconstruction is compared to other 3D
dust density reconstructions in Sec. 6. In Sec. 7 we summarize
the findings of this paper.
2. Data
We used the data from the Gaia DR2 catalogue by Brown et al.
(2018), to reconstruct the galactic dust in the nearby interstellar
−1.75 −1.50 −1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25
medium. From the Gaia data archive we extract the parallaxes,
the G-band extinction, the latitude and longitude as well as their Fig. 1: The top panel shows the natural logarithm of the ther-
respective uncertainties. A plot of the full Gaia extinction data mal dust emission map produced by the Planck collaboration
set can be seen in Fig. 1. (The Planck Collaboration et al. 2018). The color-scheme was
We select sources according to the following criteria: saturated to visually match that of the Gaia dust extinction data
shown in the middle panel. The bottom panel shows the subset
1. the above mentioned data are available for the source
of Gaia extinction data within a 600pc cube centered on the Sun;
2. the parallax ωe is inside a 600pc cube around the Sun
3. the relative parallax error is sufficiently low, ω
e/σωe > 5 the data used in this paper. The scale is natural logarithm of the
4. Priam flag 0100001 or 0100002 extinction data in magnitudes. Data points in the same direction
were averaged.
The last two criteria are suggested by Andrae et al. (2018). There
are about 3.7 million stars selected by these criteria. Fig. 1 shows
a sky average of the data points used in the reconstruction. In this
1
The version of NIFTy used for this reconstruction is available on data plot one can observe structures present also in other dust
https://gitlab.mpcdf.mpg.de/ift/NIFTy . maps, for example the Planck dust map (Fig 1).
Article number, page 2 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
3. Model is a lower bound to the uncertainty (Cramér 1946; Rao 1947)
and has been shown to be an efficient technique to take cross-
3.1. Algorithm correlations between all degrees of freedom into account
The algorithm is derived from Bayesian reasoning. In Bayesian without having to parameterize them explicitly (Knollmüller
reasoning information some data d provides about a quantity of & Enßlin 2019)
interest s is calculated according to Bayes theorem:
4. Minimize Eq. (4) with respect to m using Newton Conjugate
P(d|s)P(s)
P(s|d) = (1) gradient as second order scheme with the covariance D of
P(d) Q as curvature. The expectation value with respect to Q is
hereby approximated through a set of samples drawn from
Note that the quantity of interest can be a (possibly high-
the approximating distribution Q. Second order minimiza-
dimensional) vector, in our case it is the dust density for ev-
tion by preconditioning with the inverse Fisher metric is also
ery point in space (2563 degrees of freedom after descritization).
called natural gradient descent (Amari 1997) in the literature.
There are three main ingredients neccessary for the inference of
the quantity of interest s:
1. The likelihood P(d|s) of the data d given a realization of the
A description of the used likelihood and the prior follows.
quantity of interest s. We describe our likelihood in Sub-
sec. 3.2.
2. The prior P(s) describing the best available knowledge about
the quantity of interest s in absence of data. We describe our 3.2. Likelihood
prior in Subsec. 3.3.
3. An inference algorithm that yields a statistical summary The likelihood P(d|s) can be split into two parts, one part states
of P(s|d) given the joint distribution P(d, s) = P(d|s)P(s) how the true extinction depends on the dust density and one part
of d and s. We use the inference algorithm described in that states how the actual data is distributed given the true ex-
Knollmüller & Enßlin (2019). tinction on that line of sight. The first part, which we call the
response R, states how the unknown dust extinction density ρ
The main quantity of interest s is the logarithmic G-band imprints itself on the data. The extinction of light on the i-th line
dust extinction cross-section density s = ln(αρ/pc), henceforth of sight Li is given by the line integral
called the logarithmic dust density. Hereby ρ denotes the actual
dust mass density and α the average G-band dust cross section
per mass. The value of α is uncertain, which is why we report
extinction densities, also called dust pseudo-densities, instead. Z
(AG )i = R(ρ) i = dl α ρ(l) .
We approximate the posterior with a Gaussian (5)
Li
Q(s) = G (s − m, D) (2)
exp − 12 (s − m)† D−1 (s − m)
= 1
, (3)
|2πD| 2 Here α is the average dust cross section per unit of mass and
the line of sight Li = Li (ω) is dependent on the true parallax ω.
by adopting a suitable mean m and uncertainty dispersion D. The As noted in subsection 3.1, the value of α is uncertain
approximation is obtained by minimizing the Kullback-Leibler and we
reconstruct the dust extinction density s = ln αρ/pc instead.
divergence (Kullback & Leibler 1951)
Z The extinction is additive because the extinction data are
Q given in the magnitudes scale, which is logarithmic. The true
KL(Q, P) = dQ ln (4)
P parallax ω of the star is uncertain. We assume the true parallax
ω to be Gaussian distributed around the published parallax ω e
with respect to the parameters of Q. This approach is known with a standard deviation equal to the published parallax error
as variational Bayes (Nasrabadi 2007) or Gibbs free energy ap- σω :
proach (Enßlin & Weig 2010). The approach we take in finding
the unknown approximate posterior mean m and covariance D
of Eq. 3 is described in detail in Knollmüller & Enßlin (2019). It
can be summarized as follows:
ω, σω ) = G (ω − ω
P(ω|e e, σ2ω ) (6)
1. Calculate the negative log-probability −log(P(s, d)) for the
problem, disregarding normalization terms like the evidence
P(d).
2. Perform coordinate transformations of the unknown quan- The parallaxes of Gaia DR2 were shown to be Gaussian dis-
tities until those are a-priori Gaussian distributed with unit tributed with incredible reliability by Luri et al. (2018). How-
covariance (Kucukelbir et al. 2017; Knollmüller & Enßlin ever, it was also noted by the same authors that there can be
2018). outliers. By restricting ourselves to close-by sources for which
3. Choose the class of approximating distributions to be Gaus- G-band extinction values are published, we expect to have cut
sian with variable mean m and covariance D = (1 + Mm )−1 , out most of the outliers.
where Mm is the Fisher information metric at the current m.
Here 1 is the contribution of the prior which was transformed We do not reconstruct the actual positions of the stars in our
in step 2 to have unit covariance. This uncertainty dispersion reconstruction, thus we have to marginalize them out to obtain
Article number, page 3 of 19A&A proofs: manuscript no. paper
the response: S kk log(ρ)|S kk AG |ρ fG |AG
A
ωi , σω1 , ρ) =
P((AG )i |e
Z Fig. 2: Graphical representation of the data model for our re-
= dωi P((AG )i , ωi |e ωi , σωi , ρ) construction. The logarithmic dust density log(ρ) is a Gaus-
Z sian process with a smooth Gaussian process power spectrum
= dωi P((AG )i |ωi , ω ei , σωi , ρ)P(ωi |ω ei , σωi , ρ) S kk0 = 2πδ(k − k0 )P s (k). The true extinctions AG are directly
dependent on the dust density ρ on each line of sight. The mea-
Z
sured extinctions AfG are assumed to be distributed around the
= dωi P((AG )i |ωi , ρ)P(ωi |e ωi , σωi ) true extinctions AG following a truncated Gaussian distribution
Z Z ! as described in section 3.2.
= dωi δ (AG )i − dli αρ(li ) P(ωi |e ωi , σωi )
Li (ωi )
Z Z ! Here cdfG (R(ρ)i ,Nii ) denotes the cumulative density function of a
≈ δ (AG )i − dωi dli αρ(li ) P(ωi |e
ωi , σωi ) normal distribution with mean R(s)i and variance Nii . We took
Li (ωi ) the boundaries of the truncated Gaussian to be AGmin = 0 and
AGmax = 3.609 mag as recommended in Andrae et al. (2018).
Z Z !
= δ (AG )i − dωi dli α1[0, ω1 ] (li ) ρ(li ) G (ωi − ω
ei , σ2ωi )
Li (0)
li − ω
1
3.3. Prior
Z
ei
= δ (AG )i − dli αρ(li ) sfG .
(7)
Li (0) σωi We assume the dust density to be a positive quantity that can vary
over orders of magnitude. The dust is assumed to be spatially
Here
Z x ! correlated and statistically homogeneous and isotropic. The sta-
1 1 tistical model is constructed to be as general as possible with
sfG (x) = 1 − dt √ exp − t2 (8)
−∞ 2π 2 these two properties in mind.
To reflect the positivity and to allow variations of the dust
denotes the survival function of a standard normal distribution
density by orders of magnitude we assume the dust density ρ to
and
( be a-piori log-normal distributed with
1 for x ∈ [a, b]
1[a,b] (x) = (9) αρ = ρ0 exp(s) , (12)
0 otherwise
where s x G (s, S ) (13)
denotes the indicator function for the closed interval [a, b]. Note
that we did an approximation where we replace the true extinc- is assumed to be Gaussian distributed with Gaussian process ker-
tion by the expected extinction. As a consequence the lines of nel S . Here ρ0 = 1/1000 pc is a constant introduced to give ρ the
sight are smeared out by the parallax uncertainty in our approx- correct unit and to bring it to roughly the right order of mag-
imation. This smoothing can be regarded as a first order cor- nitude. By using an exponentiated Gaussian process we allow
rection for the uncertainty of the parallax and was already used the dust density to vary by orders of magnitude while simul-
by Vergely et al. (2001). A fully Bayesian analysis would treat taneously ensuring that it is a positive quantity. In Eq. (13) S
the true parallax as unknown and infer these along the other un- is the prior covariance. If we assume no point or direction to
knowns, but this is beyond the scope of this work. be special a-priori, then according to the Wiener-Khinchin the-
For the algorithm, the integral in Eq. (7) is discretized into orem S can be fully characterized by its spatial power spectrum
a weighted sum, such that each voxel contributes to the line in- S kk0 = 2πδ(k − k0 )P s (k). We non-parametrically infer this power
tegral over Li exactly equal to the length of the line segment spectrum P s (k) as well. There are two main motivations to re-
of Li within that voxel while being discounted by the probabil- construct the power spectrum. From a physical perspective the
ω, σω ) of that voxel being on the line of sight. Apply-
ity P(l|e power spectrum provides valuable insights into the underlying
ing the response R thus takes O(Ndata Nside ) operations, where processes. From a signal processing point of view, many linear
Ndata = 3 661 286 is the number of data points used in the re- filters can be identified with a Bayesian filter that assumes a cer-
construction and Nside = 256 is the number of voxels per axis. tain prior power spectrum. The optimal linear filter is obtained
Due to the large number of data points, evaluating the response when the power spectrum used for the filter is exactly equal to
on a computer turns out to be numerically expensive. The infer- the power spectrum of the unknown quantity (Enßlin & From-
ence algorithm (see Sec. 3.1) is a minimization for which this mert 2011). However, the power spectrum of the unknown quan-
response has to be evaluated many times. To make the dust in- tity is usually also unknown, thus one has to reconstruct it as
ference feasible in a reasonable amount of time we restricted our well. While this argumentation holds for linear filters, certainly
reconstruction to a 600pc cube centered on the Sun. many aspects of it carry over to nonlinear filters such as the re-
The second part of the likelihood states how the published construction performed in this paper.
data AfG is distributed given the true extinctions (AG )i . We use Fig. 2 depicts the hierachical Bayesian model for the extinc-
the data likelihood recommended by the Gaia collaboration in tion data A fG resulting from the logarithmic dust density ln(ρ),
Andrae et al. (2018). This likelihood assumes the data A fG to be which itself is shaped by the power spectrum P s (k).
distributed according to a truncated Gaussian with a global vari- Our statistical model for the power spectrum P s (k) is a falling
power law with Gaussian distributed slope and offset modified
ance N = 0.46 mag 2 1. This leads to the likelihood
by differentiable non-parametric deviations. It is up to minor
fG |ρ) =
Y G (A fGi − R(ρ)i , Nii ) details2 an integrated Wiener process (Doob 1953) on log-log-
P(A (10) scale.
i
cdfG (R(ρ)i ,Nii ) (AGmax ) − cdfG (R(ρ)i ,Nii ) (AGmin )
2
The amplitude model given by Eq. 14 is not exactly equivalent to
for d ∈ [AGmin , AGmax ] . (11) an integrated Wiener process, but shown by (Arras et al. 2019b) to be
Article number, page 4 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
This is realized by the following formula: kernel as well as the one assumed by Lallement et al. (2018) in
p Fig. 4. Certain biases can appear when using a fixed kernel, for
P s (k) = example introducing a characteristic length scale of the order of
a
!! the FWHM of the kernel.
exp (φm σm + m̄)log(k) + φy σy + ȳ + F−1 τ(t) (14)
log(k)t
1 + t2/t02
Here φm , φy and τ(t) are the parameters to be reconstructed,
σm = 1, m̄ = −4, σy = 2., ȳ = −16, σy = 3., a = 11, t0 = 0.2 are 1.0
fixed hyperparameters, F−1 log(k)t denotes the inverse Fourier trans-
form on log-scale, and V = (600 pc)3 is the total volume of the 0.8
reconstruction. These hyperparameters settings were determined
by trial and error such that data measured from a prior sample has
0.6
roughly the same order of magnitude as the actual data and such
that the dust density varies by more than one order of magnitude
in prior samples. 0.4
In our reconstruction the parameter τ for the smooth devia-
tions of the log-log power spectrum was discretized using 128
0.2
pixels. The mathematical motivation to take Eq. (14) as a gener-
ative prior for power spectra is discussed in Arras et al. (2019b).
As a rule of thumb, k-modes for which the data constrains the 0.0
power spectrum very well will be recovered in great detail due to 0 20 40 60 80 100 120 140
the non-parametric nature of the model. For k-modes on which pc
the data provide little information, the power spectrum will be
complemented by the prior which forces it into a falling power Fig. 4: The log-normal process normalized 2-point correlation
law whenever the data is not informative. If the actual physi- reconstructed by our method (solid line) and imposed in the re-
cal process deviates strongly from a falling power law for the construction by Lallement et al. (2018) (dashed line). One can
unobserved k-modes, the prior might artificially supress or am- see that the dust is assumed to be strongly correlated at a distance
plify the posterior uncertainty of the result, possibly biasing the scale of up to about 30pc. This plot shows normalized one di-
uncertainty quantification. Fig. 3 shows a few examples of prior mensional cuts through the three dimensional Fourier transform
samples of power spectra using our choice of hyperparameters. of the log-normal spatial correlation power spectrum shown in
While the individual samples might not look too different qual- Fig. 7.
itatively, it should be noted on the one hand that any kind of
power spectrum is representable with our model given enough
data and on the other hand that the figure depicts the power spec- Putting together likelihood and prior, the overall joint infor-
trum of the log-density on log-log scale. A small deviation in mation hamiltonian for our parameters ξ, τ, and φ is
this figure can have a huge impact on the actual statistics. Re-
P(d, ξ, τ, φ) = TGAGmin , AGmin , 0.46, d (R(αρ))
G ((ξ, φ, τ)T , 1) (15)
Z
where R(αρ) i = dl α ρ(l)
(16)
104 Li
1
αρ =
p
102 exp F−1
xk P s (k)V(φ, τ) ξk (17)
1000
100
10−2 p
−4
P s (k)(φ, τ) =
10 !!
a
10−6 exp (φm σm + m̄)log(k) + φy σy + ȳ + F−1 τt (18)
1 + t2/t02
log(k)t
10−8 Y G ( x̄i − xi , σ2 )
TG xmin , xmax , σ, x̄ (x) =
10−10 i
cdfG (xi ,σ2 ) (xmax ) − cdfG (xi ,σ2 ) (xmin )
10−2 10−1
(19)
1/pc
for x ∈ [xmin , xmax ],
Fig. 3: Several prior samples of the logarithmic spatial correla-
tion power spectrum in units of pc3 . is a truncated Gaussian, and V = (600 pc)3 . The application, cal-
culation of the gradient, and the application of the Fisher met-
ric of Eq. (15) scales almost linearly with the number of voxels
constructing the power spectrum is equivalent to reconstructing 3
Nside , more specifically it takes O(Ndata Nside +Nside
3
logNside ) oper-
the correlation kernel. We show our reconstructed normalized ations to evaluate Eq. 15, where Ndata = 3 661 286 is the number
equivalent to it in a certain limit while still allowing a numerically stable of data points used in the reconstruction and Nside = 256 is the
transformation of the prior to a white Gaussian. number of voxels per axis.
Article number, page 5 of 19A&A proofs: manuscript no. paper
4. Simulated Data Test within the recovered approximate posterior uncertainty, apart
from outliers which make up about 0.15% of the voxels. See
4.1. Data generation Fig. 6 for a histogram of the uncertainty weighted residual.
In this section, a test on simulated data is presented. This test Overall, the reconstruction seems very reliable on a qualita-
enables comparing the results of the reconstruction to a known tive and quantitatively level within the unceratinty for most of
ground truth. As ground truth dust density public data from the the voxels.
SILCC colaboration (Walch et al. 2015) was used, more specif-
ically from the magneto-hydrodynamic simulation of the inter-
stellar medium B6-1pc at 50 Myr published by Girichidis et al. 5. Results from Gaia Data
(2018). This simulation result spans a cube with size (512 pc)3 . We reconstruct the dust density in a 600pc cube using 2563 vox-
We computed our synthetic ground truth differential absortion els, resulting in a resolution of (2.34 pc)3 per voxel. For our re-
ρmock from the gas density of the simulation ρsim via constructed volume we also infer the spatial correlation power
s ! spectrum of the log-density, see Fig. 7.
512x cm3 1 In Fig. 8b one can see a projection of the reconstructed dust
ρmock x − (150, 150, 0) = ρsim
T
. (20)
3
1017
600 g pc onto the sky in galactic coordinates. Fig. 9b shows the corre-
sponding expected logarithmic dust density.
Thus we stretch the 512 pc simulated cube to the 600 pc of our In Fig. 8a the projection of the dust reconstruction on the
reconstruction, scale it with a constant factor, and shift it by galactic plane is shown. This view is especially interesting to
150 pc. The shift is performed in order to have an underdense study the dust morphology as this projection introduces no
region at the center. We also take the third root of the gas density perspective-based distortion. It is especially suited to spot un-
in Eq.(20). There are two reasons for this. derdense regions such as the local bubble in high resolution. A
A practical motivation for taking the third root is that it re- logarithmic plot of the projection on the galactic plane can be
duces the dynamic range. If one does not do this, the sky will be seen in Fig. 9a. We show integrated dust density for sightlines
dominated by one very small, but very strongly absorbing blob. parrallel to the x-, y-, and z-axis in Fig. 13.
A more physical motivation is that very dense regions lead to We provide posterior uncertainty estimation via samples.
star formation. These forming stars again reduce the density by One should note that these uncertainties might be underesti-
blowing the material out of these regions. This feedback mech- mated due to the variational approach taken in this paper. One
anism was not included into the simulation by Girichidis et al. can see a map of the expected posterior variance of the sky pro-
(2018) but was shown to have a strong impact on the gas density jection in Fig. 10a and in the plane projection in Fig. 11a.
in a followup simulation by Haid et al. (2018). The third root
can be seen as a very crude way of reducing the density in these
overdense regions. 5.1. Using the reconstruction
To obtain the synthetic data from the ground truth differential The results of the reconstruction are provided online on https:
extinction cube ρmock , the following operations were performed: //wwwmpa.mpa-garching.mpg.de/~ensslin/research/
1. Sampling ground truth parallaxes ωi x G (ωi − ω ei , σ2i ) ac- data/dust.html, or by its doi:10.5281/zenodo.2577337, and
cording to the parallax likelihood published by the Gaia cool- can be used under the terms of the ODC-By 1.0 license. Proper
laboration. attribution should be given to this paper as well as to the Gaia
2. Integrating the dust density from the center of the cube to collaboration (Gaia Collaboration et al 2016).
the location of the sampled star location 1/ω using the full We give an overview of known systematic effects and advice
resolution of 5123 voxels3 . on how to use the provided dust map.
3. Sampling an observed extinction magnitude according to the
truncated Gaussian likelihood described in section 3.2. – We do not recommend to use the outer 15pc of the re-
construction. Periodic boundary conditions were assumed
for algorithmic reasons, which leads to correlations leaking
4.2. Results around the border of the cube. The inferred prior correlation
kernel (Fig. 4) suggests that correlations are vanishing after
We were able to recover a slightly smeared out version of the
30pc.
original synthetic extinction cube. In Fig. 5 integrations with re-
– We provide posterior samples. When doing further analysis
spect to the x-, y-, and z-axis of the synthetic extinction cube
of our reconstruction we recommend doing so for every sam-
and the reconstructed extinction cube are shown. This visually
ple in order to propagate errors.
confirms the reliability of the reconstruction.
– It was observed in Sec.4.2 that there can be a small number
For a more quantitative analysis, we compared the recon-
of outliers, that is differential dust extinction values that are
structed differential extinctions with the ground truth voxel-wise.
much larger than the reconstructed value, by amounts that
More specifically, we computed the uncertainty weighted resid-
cannot be explained by the reconstructed unceratinty.
ual r
– We anticipate a perception threshold that leads to the absence
ρreconstruction − ρground truth of extremely low density dust clouds. The two main reasons
r= , (21)
σρ for this are that the truncated Gaussian likelihood provides
less evidence in the regime where the extinction is close
where ρreconstruction and σρ are the posterior mean and stan- to zero and that variational Bayesian schemes are known
dard deviation computed from the approximate posterior sam- to underestimate errors. Studying a larger volume will shed
ples. The ground truth differential extinction was recovered well further light on this subject, as sightlines for more distant
3
Note that the simulation of which the data is used was performed on stars still provide information about nearby dust clouds. One
an adaptive grid. The full resolution of 5123 is only realized in the high should note that the overall Gaia extinction data provides 20
density regions. times more sightlines than were used in this reconstruction.
Article number, page 6 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
6. Discussion
Here we discuss qualitative, quantitative, and methodological
differences to other dust mapping efforts. In table 1 a detailed
break down of methodological differerences to other papers are
shown. There are three notable differences of our method to other
methods that we would like to stress.
1. The here used dataset is one of largest one used so far.
2. We use a high amount of data while still taking 3D correla-
tions into account.
3. We reconstruct the spatial correlation power spectrum. The
motivation and impact of this already briefly discussed in
Sec. 3.3
Article number, page 7 of 19A&A proofs: manuscript no. paper
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(c) (d)
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0 0
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0.50 0.50
−300 −300
−300 −200 −100 0 100 200 300 −300 −200 −100 0 100 200 300
(e) (f)
Fig. 5: Results of our test using simulated data. The rows show integrated dust extinction for sightlines parallel to the z- x- and y-
axis respectively. The first column corresponds to the test reconstruction, the second column is the ground truth synthetic extinction.
Article number, page 8 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
0.5
0.4
0.3
0.2
0.1
0.0
−10 −8 −6 −4 −2 0 2 4 6
Fig. 6: The gray curve shows a normalized histogram of the de-
viation of the reconstruction from the true solution, in sigmas.
The black curve is the probability density function of a standard
normal distribution, which is plotted as a reference. Note that the
values were clipped to the range from −10 to 10, i.e. the bump
in the gray curve at −10 corresponds to outliers that can be up
to 250 sigmas. These outliers correspond to about 0.15% of all
voxels.
107
106
105
104
103
102
101
100
10−1
10−2 10−1
1/pc
Fig. 7: The log-normal process spatial correlation power spec-
trum inferred in our reconstruction (solid line) as well as the im-
posed power spectrum of Lallement et al. (2018) (dashed line).
The shaded area around the solid line indicates 1σ error bounds.
The unit of the y-axis is pc3 . The functions can be interpreted as
the a-priori expected value of |Fln(ρ)|2 /V, where V is the vol-
ume the density ρ is defined on and F is the Fourier transform.
The region between 0.0008/pc and 0.426/pc is almost power-law
like with a slope of 3.1, the spectral index of the power law.
Article number, page 9 of 19A&A proofs: manuscript no. paper
300 2.00
1.75
200
1.50
100
1.25
0 1.00
0.75
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−300 −200 −100 0 100 200 300 0.5 1.0 1.5 2.0
(a) (b)
300 2.00
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100
1.25
0 1.00
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0.50
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−300 −200 −100 0 100 200 300 0.5 1.0 1.5 2.0
(c) (d)
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200
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0
1.00
−100 0.75
0.50
−200
0.25
−300
−300 −200 −100 0 100 200 300 0.5 1.0 1.5 2.0
(e) (f)
Fig. 8: The left column shows integrated dust extinction from −300 pc to 300 pc for sightlines perpendicular to the galactic plane.
The image covers a 600pc cube centered around the Sun. The units are e-folds of extinction. The Sun is at coordinate (0, 0), the
galactic center is located towards the bottom of the plot, and the galactic West is on the left. The right column shows all-sky
integrated dust extinction maps of the same region, but for sightlines towards the location of the Sun. The first row is the result of
the reconstruction discussed in this paper, the second row is the reconstruction performed by Lallement et al. (2018), the last row
shows the reconstruction by Green et al. (2018).
Article number, page 10 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
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0
200
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(a) (b)
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(e) (f)
Fig. 9: A natural logarithmic version of Fig. 8.
Article number, page 11 of 19A&A proofs: manuscript no. paper
0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 1.0
(a) (b)
0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 1.0
(c) (d)
Fig. 10: Uncertainty of the reconstruction of this paper derived from posterior samples (first row) and of the reconstruction of Green
et al. (2018) (second row), both in the sky projection. The uncertainties are in the same unit as the corresponding maps in Fig. 8,
or dimensionless for logarithmic uncertainties. The first column shows the variance for the dust extinction and the second column
shows the variance of the logarithmic projected dust density on natural log-scale, which can be interpreted as a relative error.
Article number, page 12 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
300 0.25 300 0.8
0.7
200 200
0.20
0.6
100 100
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0 0 0.4
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0.1
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(a) (b)
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0.2
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0.1
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−300 −200 −100 0 100 200 300 −300 −200 −100 0 100 200 300
(c) (d)
Fig. 11: Posterior uncertainty of the reconstruction of this paper derived from samples (first row) and of the reconstruction of Green
et al. (2018) (second row) in the plane projection. The uncertainties are in the same unit as the corresponding maps in Fig. 8, or
dimensionless for logarithmic uncertainties. The first column shows the variance for the dust extinction and the second column
shows the variance of the logarithmic projected dust density on natural log-scale which can be interpreted as a relative error.
Article number, page 13 of 19this paper Sale & Magorrian (2018) Kh et al. (2018b) Lallement et al. (2018) Green et al. (2018)
parallax uncertainty smoothing
√ only marginalization by sampling neglected neglected
√ proper uncertainty handling
max distance 300 3 pc 5 kpc 6 kpc ≈ 2 2 kpc 3 kpc
max voxel resolution 2.3 pc not applicable about 200 pc 5 pc 16.4 pc/0.063 pc
number of datapoints 3.7 million 6 349 21 000 71 357 806 million
power spectrum inference yes no no no no
correlations 3D 3D 2D map only 3D 1D correlations only
positiveness yes only of reddening no yes yes
statistical method Variational Bayes Expectation Propagation analytic maximum posterior Hamiltonian Monte Carlo
data sets Gaia DR2 synthetic Gaia data APOGEE Gaia DR1 + APOGEE + 2MASS Pan-STARRS + 2MASS
Table 1: A table comparing different dust inference methods with the one performed in this paper. The first row indicates how the parallax uncertainty of the stars was treated.
Hereby smoothing refers to weighting a voxel in the line of sight by the survival function of the star radial distance, as is described in Eq. (7). The distance of the furthest point
in the reconstruction is given in the second row. The dimensions of the smallest voxel are given in the third row. For the reconstruction of Sale & Magorrian (2018) the concept
of voxel resolution is not readily applicable; Sale & Magorrian (2018) use 140 inducing points spanning a region for which one could evaluate the posterior mean at any point.
The resolution for Green et al. (2018) contains two values because the resolution is different in radial/angular direction. The fourth row provides the number of used data points.
The fifth row indicates whether the power spectrum is inferred. The sixth row states which kind of correlations are assumed for the reconstruction. Whether positivity of dust
density is enforced can be read in the seventh row. The second to last row states the method, with which the posterior summary statistics was calculated from the unnormalized
log posterior. In the last row the data sets used for the reconstruction are listed.R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
We compare our dust map to other maps. Comparisons to power spectra agree more or less for the larger modes (low k),
2D dust maps are only possible on a qualitative level, since it is where the data is very constraining.
not clear whether structures visible in the 2D maps that are not One can empirically compute power spectra of the dust den-
present in the 3D map are simply further away or are too noisy sity using a Fourier transformation. A comparison plot with all
in the data for the algorithm to pick them up. On a qualitative the three mentioned reconstruction can be found in Fig. 14. This
level it is possible to see several morphological similarities of shows a white noise floor in the reconstruction of Green et al.
our reconstruction in Fig. 8b to the Planck dust map (The Planck (2018), which can visually also be seen as small scale structures
Collaboration et al. 2018) in Fig. 1. These figures also show that in the plane projections shown in Figs. 8 and 9.
many dust structures that are not inside the galactic plane are
local features.
The two 3D dust maps mentioned in Sec. 1 permit a more 7. Conclusions
thorough analysis. Fig. 8 shows a compilation of projected dust 1. We provide a highly resolved map of the local dust density
densities for our reconstruction as well as the reconstruction of using only Gaia data. This map agrees on large scales with
Lallement et al. (2018) and Green et al. (2018)4 , restricted to previously published maps of Lallement et al. (2018) and
the same volume as the reconstruction discussed in this paper. A Green et al. (2018), but also shows significant differences on
logarithmic version of this figure is provided by Fig. 9. small scales. These differences might to a large degree stem
While our map seems to agree on large scales with the other from the different data used. Our map shows many structures
maps, there seems to be a pronounced tension in the predictions visible in the Planck dust map (The Planck Collaboration et
of the position of some dust clouds compared to the reconstruc- al. 2018).
tion of Lallement et al. (2018). Compared to the map of Green 2. In comparison to previous maps, we were able to improve on
et al. (2018) we recover the small scales significantly better and 3D resolution while still being mostly consistent on the large
suffer far less from radial smearing. It should be noted that Green scales. A comparison to 2D maps like the Planck dust map
et al. (2018) maped a significantly larger part of our galaxy, and seems to confirm the features present in our map.
that the region that overlaps with our map was declared to be 3. We find that the logarithmic density of dust exhibits a power-
not that reliable by the authors themselves. The differences are law power spectrum with a 3D spectral index of 3.1, corre-
probably due to the different nature of the used datasets. The sponding to a 1D index of 1.1. This is a significantly harder
Gaia DR2 data used in our reconstruction has a vastly higher spectrum as that expected for a passive tracer in Kolmogorov
amount of data points than those used for the other reconstruc- turbulence, which would be a 1D index of 5/3. The harder
tions. These data points, taken from Gaia DR2, have a very small spectrum is probably caused by the sharp edges of the local
parallax error. Additionally our reconstruction takes the full 3D bubble and other ionization or dust evaporation fronts.
correlation structure into account. 4. In contrast to other dust reconstructions, we predict very low
Our reconstruction as well as the reconstruction of Green dust densities inside the local bubble. This discrepancy is
et al. (2018) permit quantifying uncertainties using samples. A possibly an artifact of our reconstruction as there are known
plot of uncertainties of the dust density reconstructions projected dust clouds in our vicinity, for example Barnard 68 (Alves
into the galactic plane can be seen in Fig. 11. Uncertainties of the et al. 2001) and the local Leo cold cloud (Peek et al. 2011).
dust density reconstructions in the sky projection can be seen in These examples are however considerably smaller than a
Fig. 10. voxel of our simulation and we cannot exclude that the re-
To quantify the dynamic range of the reconstruction and as gion in our galactic vicinity is actually as void of dust as it
a prediction on the variability of the logarithmic dust density is in our reconstruction when disregarding structures smaller
we calculated histograms of dust density which show how many than 3 pc. The possibility that Gaia extinction estimates are
voxels have which dust density. These histograms can be seen biased for small distances can also not be excluded.
in Fig. 12a. One can see that the histogram of our reconstruc- 5. Our reconstruction provides further evidence that the North-
tion extends slightly more towards high dust densities and sub- ern Galactic Spur is at distances greater than 300 pc, as was
stantially towards low dust densities. This is possibly because already found by Lallement et al. (2016).
our reconstruction is more sharply resolved, thus regions of high 6. We hope that by providing accurate reconstructions of the
dust density get captured better and bleed less into the regions nearby dust clouds, further studies of dust morphology will
where dust is absent. be possible as well as the construction of more accurate ex-
We characterize how much pairs of those reconstructions tinction models for photon observations in a large range of
agree by the heatmaps of their voxel-wise value pairs. These frequency bands.
heatmaps can be seen in Fig. 12. For two perfectly agreeing re- Acknowledgements. We acknowledge helpful commentaries and suggestion
constructions the heatmap would show a line with slope 1. Again from our anonymous referee as well as fruitful discussions with S. Hutschen-
it can be seen that the dust density in our reconstruction varies reuter, J. Knollmüller, P. Arras and others from the information field theory
significantly more than in the two other reconstruction. While group at the MPI for astrophysics, Garching. This work has made use of
data from the European Space Agency (ESA) mission Gaia (https://www.
all maps agree more or less for high dust densities, our dust map cosmos.esa.int/gaia), processed by the Gaia Data Processing and Anal-
exhibits vastly more volume with low dust density. ysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/
The reconstruction of Lallement et al. (2018) is performed consortium). Funding for the DPAC has been provided by national institutions,
in particular the institutions participating in the Gaia Multilateral Agreement.
using Gaussian process regression, as is ours. Thus one can com-
pute the prior Gaussian process correlation power spectrum used
in their reconstruction. Fig. 7 shows both our inferred power
spectrum as well as their assumed power spectrum. These two References
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R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
10−1
103
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−4
Lallement et al
10 −3 102
−6
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this paper
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Green et al
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this paper
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(c) (d)
Fig. 12: Panel a shows normalized histograms of dust densities. The solid line corresponds to our reconstruction, the dashed line
is the reconstruction of Lallement et al. (2018) and the dash-dotted line is the reconstruction of Green et al. (2018). The other three
plots are heatmaps of voxel-wise correlations between reconstructed logarithmic dust densities, where the color shows bin counts.
The black line in the heatmaps is the identity function, corresponding to perfect correlation.
Article number, page 17 of 19A&A proofs: manuscript no. paper
300 2.00 300
0
1.75
200 200
1.50 −1
100 100
1.25
−2
0 1.00 0
−3
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(a) (b)
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(e) (f)
Fig. 13: The reconstructed dust density in different projections. The rows show integrated dust extinction for sightlines parallel to
the z- x- and y- axis respectively. In the first rows, the galactic center is located towards the bottom of the plot, in the other two
rows the galactic North is located towards the top of the plot. The cube is in galactic coordiantes, thus the x-axis is oriented towards
the galactic center and the z-axis is perpendicular to the galactic plane. The first column shows the integrated G-band extinction in
e-folds of extinction, the second column is a logarithmic version of the first column.
Article number, page 18 of 19R. H. Leike and T. A. Enßlin : Charting nearby dust clouds using Gaia data only
102
100
10−2
10−4
10−6
10−8
10−2 10−1
1/pc
Fig. 14: Empirical spatial correlation power spectra of the re-
constructed mean dust density in units of pc. The black line was
computed from our reconstruction, the dark-grey line is com-
puted from the reconstruction of Lallement et al. (2018) and
the light-grey line is computed from the reconstruction of Green
et al. (2018). For the reconstruction of Green et al. (2018) un-
specified voxel values on sightlines that lacked data were re-
placed with 0.
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